Table Of ContentWeak selection and stability of localized distributions in Ostwald ripening
Boaz Giron, Baruch Meerson and PavelV. Sasorov∗
The Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel
Wesupport andgeneralize a weak selection rule predictedrecently for theself-similar asymptotics
8 of the distribution function (DF) in the zero-volume-fraction limit of Ostwald ripening (OR). An
9
asymptoticperturbationtheoryisdevelopedthat,whencombinedwithanexactinvarianceproperty
9
of the system, yields the selection rule, predicts a power-law convergence towards the selected self-
1
similarDFandagreeswellwithournumericalsimulationsfortheinterface-anddiffusion-controlled
n OR.
a
J PACS numbers: 05.70.Fh, 64.60.-i, 47.54.+r
8
1
In a late stage of a first-order phase transition a two- Theself-similarityandrelatedscalingswerediscovered
] phasemixtureundergoescoarsening,orOstwaldripening byLS[1,2]inthecaseofn=1(diffusion-controlledOR).
h
(OR), when the minority phase tends to minimize its LS arrived at a unique self-similar DF (we will call it
c
e interfacial energy under condition of a constant volume the limiting solution), and ruled out other possible so-
m [1–3]. Despite numerous works, OR continues to attract lutions. In the first paper [1], the other solutions were
- attentionbothinexperiment[4]andintheory[5,6]. Our rejected as non-normalizable with respect to Eq. (2).
t
a main motivation in studying this problem has been an In the case of n = 0 (interface-controlled OR) this ar-
t attempt to resolve an old selection problem (described gument was repeated by Wagner [3]. However, already
s
. below) that created much controversy. in their second paper [2] on the same subject LS real-
t
a The“classical”formulationoftheproblemofOR,valid ized that no problem with normalization arises for ini-
m
in the limit of a negligibly small volume fraction of the tially localized DFs (that is, for those with a compact
- minority domains, is due to Lifshitz and Slyozov (LS) support at t = 0). This correction was apparently over-
d
[1,2] and Wagner [3]. In this formulation, the dynamics looked in the literature (e.g., Ref. [7]), until Brown [8]
n
o of the distribution function (DF) F(R,t) of the domain addressed the other solutions and found them numeri-
c sizes are governed(in scaled variables) by the continuity cally for n = 1. This created a long-standing contro-
[
equation versy (see, e.g. [9]), and the first step towards resolving
1 it was made in the case of n = 0 [6]. It was noted that
∂F ∂ 1 1 1
v + (VF)=0, V(R,t)= − , (1) a DF, initially localized on an interval [0, R (t = 0)],
8 ∂t ∂R Rn (cid:18)Rc R(cid:19) always remains localized on a (time-dependenmt) interval
7
whereR (t)isthecriticalradiusforexpansion/shrinkage [0, R (t)]. Furthermore,if F(R, t=0)is describableby
1 c m
ofan individualdrop, whilen is determined by the mass a power law A [R (t=0)−R]λ in the close vicinity of
1 0 m
0 transfer mechanism. The dynamics are constrained by R = Rm(t = 0), then for any t > 0 the leading term in
8 conservationofthetotalvolumeoftheminoritydomains: theexpansionofF(R, t)inthevicinityofR=R (t)has
m
9 ∞ the form A(t)[Rm(t)−R]λ. Invariance of the exponent
t/ R3F(R,t)dR =Q=const. (2) λunder the dynamics(1)and(2)implies a(weak)selec-
a Z
0 tion rule for the “correct” self-similar DF, as there is a
m
Of great interest are possible self-similar intermedi- one-to-onecorrespondencebetween0<λ<+∞andthe
-
d ate asymptotics of this problem and the rule that se- parameterσ [6]. [Thelimitingsolutioncorrespondstoan
n lects the relevant asymptotics out of many possibilities. extended(non-compact)initialconditionor,formally,to
o Scaling analysis of Eqs. (1) and (2) yields a similarity λ→+∞.] Moreprecisely,ifa self-similarasymptoticsis
:c ansatz F(R,t) = t−µΦ(Rt−ν) and Rc = (t/σ)ν, where everreached,it must be the one selectedby λ. However,
v µ = 4/(n+ 2), ν = 1/(n+ 2) and σ = const. Upon no attempts have been made to solve (even numerically)
Xi substitution, one finds a family of self-similar DFs for the full time-dependent problem with a localized initial
every n ≥ −1, where each of the DFs is localized on a DF. Furthermore, no stability/convergence analysis for
r
a finiteinterval[0,um]ofthesimilarityvariableu=Rt−ν. the localized DFs has been performed, so the selection
The DFs can be parameterized by σ, and the interval of rule proposed in [6] has remained unconfirmed.
possible values of σ is determined by the requirements This Letter supports the selection rule along three di-
of the continuity of F(R,t) on the whole interval [0,um] rections. The first one is to generalize the selection rule
and normalizability with respect to Eq. (2). For each of for any n ≥ −1. The second is to prove the stability
the solutions, the average domain radius grows in time ofandanalyzethe convergencetowardsthe selectedself-
like t1/(n+2), and the number of domains decreases like similar DF. The third is to verify our theoretical predic-
t−3/(n+2), but the coefficients in these scaling laws are tions numerically.
σ-dependent.
1
A meaningful formulation of the stability problem re- Wewillseeinamomentthataself-similarsolutionfor
quires some care. Indeed, each member of the family of F(R,t)correspondstoasteady-statesolutionforG(x,τ).
self-similar solutions for the DF, except the limiting so- Therefore,wearelookingforthesolutioninthefollowing
lution,isformallyunstablewithrespecttoadditionofan form:
(infinite) tail. In this case it is the limiting solution that
willfinally develop[1,2,9]. However,sucha perturbation G(x,τ)=Φ0(x)+Φ1(x)eqτ +... ,
is not always possible. In addition, the results of [6] im- v(τ)=v +v eqτ +... , (6)
0 1
ply that each member of the family, except the limiting
solution, is formally unstable with respect to a localized where q is a (sought for) complex number. Both Φ0(x),
perturbation that either has a larger Rm than the “un- andΦ1(x)arelocalizedontheinterval[0,1]. Thepertur-
perturbed” DF, or the same R and a smaller exponent bation must not change the normalization condition (2)
m
λ. In each of these cases another self-similar solution andtheasymptoticsoftheunperturbedsolutionnearthe
from the same family finally develops (as we see in our 1
point x = 1, that is, x3Φ (x)dx = 0 and Φ = O(Φ )
numerical simulations), and this situation can hardly be 1 1 0
R0
regarded as instability. A meaningful formulation of the at x→1.
stability problem should therefore deal with initial per- A family of steady-state solutions Φ0(x) (parameter-
turbationslocalizedonthesameintervalofRasthe“un- ized by v0) is obtained from the zero-order equation
perturbed” DF,andcharacterizedby the same exponent
in the close vicinity of Rm(t=0). v0(x−n−x)+x−x−n−1 (dΦ0/dx)
Wewilldevelopanasymptoticlineartheorythat,com- + (n+1)x−n(cid:2)−2−nv x−n−1−4(v −1(cid:3)) Φ =0. (7)
0 0 0
bined with an (exact) invariance property of the model, (cid:2) (cid:3)
will enable us to prove,analytically,the stability of each Integration of this equation in elementary functions is
of the self-similar DFs. This result and our numerical possible for n=−1,0,1 and 2 (that is, for most cases of
simulations will confirm the weak selection rule [6]. We physical interest [10]). For example, for n=0 one has
will analyze the late-time convergence of an initially lo-
calizedDFtowardstheselectedself-similarDFandfinda Φ0(x)=CQx(1−x)α(x2−x)ν, 0≤x≤1, (8)
power-law decay in time for the corresponding (non-self-
where
similar)perturbation. Thisdecayismuchfasterthanthe
logarithmic decay found for the limiting solution [1,2]. 4v −5 v −5 1
0 0
α= , ν = , x = , (9)
We will see that not only the selected self-similar DF, 2−v 2−v 2 v −1
0 0 0
butalsothedecayexponentaredeterminedsolelybythe
analytical properties of F(R,t = 0) in the close vicinity 1
while C is determinedfromthe condition x3Φ (x)dx
of R (t = 0). Our theoretical predictions show a very Q 0
m R0
good agreement with simulations. =Q. Thisfamilyofsolutionsisdefinedfor5/4<v <2.
0
We will start with the asymptotic theory. Solving the It corresponds to the family of self-similar solutions for
problem analytically is made possible by a change of F(R, t) obtained in Ref. [6].
variables that employs the compactness of the support For n=1 one obtains
[0,R (t)] of the DF. Introduce a scaled drop radius and
m
a new time: Φ0 =CQx2(1−x)α(x−x−)γ2−γ1(x+−x)−γ1−γ2. (10)
t
R dt′ Here
x(R,t)= and τ = , (3)
Rm(t) Z Rmn+2(t′) 5v0−6 12−7v0 3v0
0 α= 3−2v , γ1 = 6−4v , γ2 = (6−4v )s,
0 0 0
and a scaled DF
(11)
G(x,τ)=R4 [t(τ)]F[R(x,τ),t(τ)]. (4)
m x± =(−1±s)/2,s=[(v0+3)/v0−1)]1/2 and0≤x≤1.
This family is defined for 6/5<v <3/2.
InthenewvariablesEqs. (1)and(2)canberewrittenas 0
For anyn, we will needto know the behaviorofΦ (x)
0
(∂G/∂τ)+ v(x−n−x)+x−x−n−1 (∂G/∂x) andΦ (x)intheclosevicinityofx=1. Asimpleanalysis
1
+ (n+1)x(cid:2)−n−2−nvx−n−1−4(v−(cid:3)1) G=0 (5) of Eq. (7) yields Φ0 =H0(ζ)ζα, where ζ =1−x, H0(ζ)
is an analytic function on the interval [0,1], H (0) 6= 0,
(cid:2) (cid:3) 0
1 and
and G(x,τ)x3dx = Q, respectively, where v(τ) =
R0 α= (n+4)v0−n−5. (12)
R [t(τ)]/R [t(τ)]. The function G(x,τ) is nonzero on
m c n+2−(n+1)v
0
the interval 0<x<1 and zero elsewhere.
2
The solution for Φ (x) exists if 0 < α < ∞, that is Under the transformation (3), (4) the variables R and
0
(n+5)/(n+4) < v < (n+2)/(n+1). This interval F are multiplied by some quantities independent of R.
0
of permitted values of v is non-empty for any n ≥ −1. Therefore, we can rewrite Eqs. (16) and (17) in the new
0
[The case of n=−1 is the simplest: H (ζ)=const.] variables x and τ as follows. The initial DF is now
0
Now we go to the first order in Eq. (5):
G (x)=ζλg′(ζ)+ζλ2g′(ζ), ζ >0, (18)
0 1 1 2
v (x−n−x)+x−x−n−1 (dΦ /dx)
0 1 where g′(ζ) and g′(ζ) are analytic at ζ =0, g′ (0)6=0,
+ q−(n+(cid:2)1)x−n−2−nv0x−n−1−4(v(cid:3)0−1) Φ1 and we1remind th2at ζ = 1 − x. Correspond1i,n2gly, the
(cid:2)=v (x−x−n)(dΦ /dx)+(4+nx−n−1)Φ(cid:3) . (13) time-dependent DF G(x,τ) can be written as
1 0 0
(cid:2) (cid:3) G(x,τ)=ζλ1h′(ζ,τ)+ζλ2h′(ζ,τ), ζ >0, (19)
For a given v1, this linear equation can be solved in 1 2
quadratures [11]. We will need only the leading asymp- where h′(ζ,τ) and h′(ζ,τ) are analytic functions of ζ in
totics of the solution in the close vicinity of x=1, so we ζ =0, a1nd h′ (0,τ)26=0.
write down the solution as 1,2
Theexponentsλ andλ ,prescribedbytheinitialcon-
1 2
ditions, remain invariant. Hence, the long-time asymp-
Φ =ζαχ (ζ) +ζβχ (ζ), (14)
1 1 2 totics of Eq. (19) should coincide with that given by
where β = α−q[n+2−(n+1)v ]−1, χ and χ are Eqs. (6) and (14). A direct comparison yields α = λ1
0 1 2
andβ =λ . Thefirstequalityisnothingbuta(weak)se-
analytic functions on the interval [0,1] and χ (0) 6= 0. 2
1,2
lection rule for the self-similar solution, and the selected
The solution exists if Reβ ≥ 0 which implies Req <
value of v is
(n+4)v −n−5. One can check a posteriori that this 0
0
inequality holds. (n+2)λ +n+5
1
Eq. (14) will be used later. At this stage we notice v0 = . (20)
(n+1)λ +n+4
1
that the still undetermined “eigenvalue” q must be se-
lected by the initial condition. To make this selection The second equality determines q:
possible, we should exploit an (exact) invariance prop-
3(λ −λ )
erty of Eq. (1). Consider the initial value problem −q = 2 1 . (21)
(n+1)λ +n+4
dR/dt = V(R,t),R(0) = R that describes the char- 1
0
acteristics of Eq. (1). If the solution of this problem, One can see that −q is real and positive which means
R(t;R0), is known, the solution of Eq. (1) can be writ- stability.
ten in the following form: Goingbacktothe“physical”variablesRandtiseasy.
Indeed, evaluating R (t) on the self-similar solution, we
m
F(R,t)=F0[R0(R,t)]∂R0(R,t)/∂R, (15) obtain R (t) = [(n+2)(v −1)t]1/(n+2). Then, using
m 0
Eq. (3), we see that eqτ =t−Γ, a power-lawdecay in the
whereR (R,t)isthefunctioninversetoR(t,R )withre-
0 0
“physical” time. Here
spect to the argument R . Now, R(t,R ) is an analytic
0 0
and monotonic function of R0. Therefore, the inverse 3(λ2−λ1)
function R (R,t) is also an analytic function of R, and Γ= >0.
0 (n+2)(λ +1)
1
soF(R,t)preservesitsanalyticformalongthecharacter-
istics R = R(t;R ), including the “edge” characteristics If we limit ourselves to an important particular case
0
R (t). of a single “non-trivial” exponent in the initial DF,
m
Weassumeapower-lawbehaviorofF(R, t=0)inthe G0(x) = ζλg(ζ), [where g(ζ) is analytic and g(0) 6= 0],
close vicinity of R = R (0). More precisely, we assume then G(x,τ) = ζλh(ζ,τ), where h(ζ,τ) is an analytic
m
that functions of ζ and h(0,τ)6= 0. Now, using Eqs. (6) and
(14), we obtain
F (ξ)=ξλ1g (ξ)+ξλ2g (ξ), (16)
0 1 2
3
−q = , (22)
where ξ =R (0)−R >0. Here λ and λ > λ are ar- (n+1)λ+n+4
m 1 2 1
bitrary positive numbers, such that λ −λ 6= 1, 2, ...,
2 1 unlessthelineartermintheTaylorseriesofg(ζ)atζ =0
and g (ξ) and g (ξ) are analytic at ξ = 0 such that
1 2 vanishes [12]. This yields the power exponent
g (0) 6= 0. In view of of the analyticity property men-
1,2
tioned above, Eq. (15) can be rewritten as 3
Γ= . (23)
(n+2)(λ+1)
F(R,t)=(ξ′)λ1h (ξ′,t)+(ξ′)λ2h (ξ′,t), (17)
1 2
In the limit of λ → ∞, we obtain Γ → 0. Clearly, it
where ξ′ = R (t)−R > 0, while h (ξ′,t) and h (ξ′,t) corresponds to the logarithmically slow decay obtained
m 1 2
are analytic functions of ξ′ at ξ′ =0, and h (0,t)6=0. for the limiting solution [1,2].
1,2
3
Therefore,boththeself-similarDF,andthepower-law
decayrateofasmallperturbationarounditareuniquely 1.00
determined by the asymptotics of the initial DF in the 0.90
close vicinity of the maximum domain size, R=R (0).
m
Γ 0.80
We verified the theory (in the cases n = 0 and 1) by
performingextensive numericalsimulationswith Eq. (1) 0.70
and an explicit equation for R˙ that follows from Eqs.
c FIG. 20..60 Convergence exponents predicted analytically
0.5 0.7 0.9 1.1
(1) and (2). As the dynamics is extremely sensitive to (line) and found numerically (squaλres) for different λ.
smallchangesinthevicinityofR=R (t),weneededan
m
algorithmthatpreservedthe compactnessofthe DFand
kept a high accuracy near the edge point R=R (t). A
m
simple andefficientLagrangianalgorithmwasdeveloped
[13] that satisfied these requirements. Typical simula-
tion results for the interface-controlled OR, n = 0, are
presented in Figs. 1 and 2. Figure 1 shows convergence
of an initially localized DF, F(R,0) = R(5−R)λ with ∗ On leave from the Institute of Theoretical and Experi-
λ=1,towardstheselectedself-similarDF(8),forwhich mental Physics, Moscow 117259, Russia.
[1] I.M. Lifshitz and V.V. Slyozov, Zh. Exp. Teor. Fiz. 35,
Eq. (20)predictsv =7/5. The insetshowsconvergence
0 479 (1958) [Sov. Phys.JETP 8, 331 (1959)].
of v(t) towards v = 7/5. The convergence exponent
0 [2] I.M. Lifshitz and V.V. Slyozov, J. Phys. Chem. Solids
Γexp =0.76 found numerically agrees very well with our 19, 35 (1961).
theoretical prediction Γth = 0.75. Figure 2 shows the [3] C. Wagner, Z. Electrochem. 65, 581 (1961).
convergence exponents Γexp found numerically for dif- [4] O. Krichevsky and J. Stavans, Phys. Rev. E 52, 1818
ferent λ. A good agreement with the theoretical curve (1995); W. Theis, N.C. Bartelt, and R.M. Tromp, Phys.
Γth =3/[2(λ+1)]isseen. Wealsoobservedagoodagree- Rev.Lett.75,3328 (1995); J.-M.Wenet al.,Phys.Rev.
ment between the theory and simulations in the case of Lett. 76, 652 (1996); K. Morgenstern, G. Rosenfeld and
the diffusion-controlled OR, n=1. G.Comsa,Phys.Rev.Lett.76,2113(1996);G.R.Carlow
andM.Zinke-Allmang,Phys.Rev.Lett.78,4601(1997);
J.B. Hannon et al.,Phys. Rev.Lett.79, 2506 (1997).
[5] M.Marder,Phys.Rev.A36,858(1987);J.H.Yaoetal.,
Phys. Rev. B 47, 14110 (1993); A.J. Bray, Adv. Phys.
43, 357 (1994); N. Akaiwa and P.W. Voorhees, Phys.
-1.0
Rev. E 49, 3860 (1994); V.G. Karpov, Phys. Rev. Lett.
30.0 -2.0
ln[v(t)-v0] ---543...000 7Pan4hd,ys3M.1R8.5Ge;vr.7aE5n,t,52P47,0h2yRs1(.13R9(9e1v59).9;E6N);5.6CA,.kRSa2aiw1gua(i1,a9nD9d7.S)D...OI.’GMoerimroann,
20.0 -6.00.0 2.0 4.0 6.0 [6] B. Meerson and P.V. Sasorov, Phys. Rev. E 53, 3491
x,t) ln(t) (1996).
(
G a bc [7] E.M. Lifshitz and L.P. Pitaevsky, Physical Kinetics
(Pergamon, London, 1981).
10.0 [8] L.C. Brown, Acta Metall. 37, 71 (1989); Scripta Metall.
Mater. 24, 963; 2231 (1990).
[9] M.K.ChenandP.W.Voorhees,ModellingSimul.Mater.
Sci. Eng. 1, 591 (1993).
0.0
FIG. 10..0 Conve0r.g2ence of0.a4n initia0ll.y6localiz0e.d8DF w1it.h0 [10] V.V. Slyozov and V.V. Sagalovich, Sov. Phys. Usp. 30,
23 (1987); M. Zinke-Allmang, L.C. Feldman, and M.H.
λ=1 towards the selected self-simxilar DF (8) with v0 =7/5
Grabow, Surf.Sci. Rep. 16, 377 (1992).
(solid line). Numerical solutions are shown by dotted lines
at time moments t=20 (a), 100 (b), 500 (c) and 1000. The [11] The solution is obtained by making the ansatz Φ1(x)=
inset shows the convergence of v(t) towards v0 =7/5. Φ0(x)f(x) and directly solving the equation for f(x).
[12] If it does vanish, an additional factor 2 appears in the
numeratorsoftherighthandsidesofEqs.(22)and(23).
We have demonstrated that only a weak selection is
Ingeneral,ifkistheorderofthefirstnon-vanishingterm
possible in the “classical” model of OR. To get a strong
in theTaylor series, an additional factor k appears.
selection rule, one obviously must go beyondthe “classi-
[13] B. Giron, Localized Distributions in the Problem of
cal” model. Ostwald Ripening, M.Sc. thesis, Hebrew University,
ThisworkwassupportedinpartbyagrantfromIsrael Jerusalem (1998).
ScienceFoundation,administeredbytheIsraelAcademy
ofSciencesandHumanities, andbythe RussianFounda-
tion for Basic Research (grant No. 96-01-01876).
4
